A constant function is a trivial example of a step function. Then there is only one interval, A0=R.{\displaystyle A_{0}=\mathbb {R} .}

The sign functionsgn⁡(x){\displaystyle \operatorname {sgn}(x)}, which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.

The Heaviside functionH(x), which is 0 for negative numbers and 1 for positive numbers, is an important step function, and is equivalent to the sign function, up to a shift and scale of range (H=(sgn+1)/2{\displaystyle H=(\operatorname {sgn} +1)/2}). It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

The integer part function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors[1] also define step functions with an infinite number of intervals.[1]

The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers.

A step function takes only a finite number of values. If the intervals Ai,{\displaystyle A_{i},}i=0,1,…,n,{\displaystyle i=0,1,\dots ,n,} in the above definition of the step function are disjoint and their union is the real line, then f(x)=αi{\displaystyle f(x)=\alpha _{i}\,} for all x∈Ai.{\displaystyle x\in A_{i}.}

The Lebesgue integral of a step function f=∑i=0nαiχAi{\displaystyle \textstyle f=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}\,} is ∫fdx=∑i=0nαiℓ(Ai),{\displaystyle \textstyle \int \!f\,dx=\sum \limits _{i=0}^{n}\alpha _{i}\ell (A_{i}),\,} where ℓ(A){\displaystyle \textstyle \ell (A)} is the length of the interval A,{\displaystyle A,} and it is assumed here that all intervals Ai{\displaystyle A_{i}} have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.[2]